TY - JOUR

T1 - Rational realization of the minimum ranks of nonnegative sign pattern matrices

AU - Fang, Wei

AU - Gao, Wei

AU - Gao, Yubin

AU - Gong, Fei

AU - Jing, Guangming

AU - Li, Zhongshan

AU - Shao, Yanling

AU - Zhang, Lihua

N1 - Publisher Copyright:
© 2016, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.

PY - 2016/9/1

Y1 - 2016/9/1

N2 - A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set {+,−, 0} ({+, 0}, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of A. Using a correspondence between sign patterns with minimum rank r ≥ 2 and point-hyperplane configurations in Rr−1 and Steinitz’s theorem on the rational realizability of 3-polytopes, it is shown that for every nonnegative sign pattern of minimum rank at most 4, the minimum rank and the rational minimum rank are equal. But there are nonnegative sign patterns with minimum rank 5 whose rational minimum rank is greater than 5. It is established that every d-polytope determines a nonnegative sign pattern with minimum rank d + 1 that has a (d + 1) × (d + 1) triangular submatrix with all diagonal entries positive. It is also shown that there are at most min{3m, 3n} zero entries in any condensed nonnegative m × n sign pattern of minimum rank 3. Some bounds on the entries of some integer matrices achieving the minimum ranks of nonnegative sign patterns with minimum rank 3 or 4 are established.

AB - A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set {+,−, 0} ({+, 0}, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of A. Using a correspondence between sign patterns with minimum rank r ≥ 2 and point-hyperplane configurations in Rr−1 and Steinitz’s theorem on the rational realizability of 3-polytopes, it is shown that for every nonnegative sign pattern of minimum rank at most 4, the minimum rank and the rational minimum rank are equal. But there are nonnegative sign patterns with minimum rank 5 whose rational minimum rank is greater than 5. It is established that every d-polytope determines a nonnegative sign pattern with minimum rank d + 1 that has a (d + 1) × (d + 1) triangular submatrix with all diagonal entries positive. It is also shown that there are at most min{3m, 3n} zero entries in any condensed nonnegative m × n sign pattern of minimum rank 3. Some bounds on the entries of some integer matrices achieving the minimum ranks of nonnegative sign patterns with minimum rank 3 or 4 are established.

KW - condensed sign pattern

KW - convex polytope

KW - integer matrix

KW - minimum rank

KW - nonnegative sign pattern

KW - point-hyperplane configuration

KW - rational minimum rank

KW - rational realization

KW - sign pattern (matrix)

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U2 - 10.1007/s10587-016-0299-1

DO - 10.1007/s10587-016-0299-1

M3 - Article

AN - SCOPUS:84991329283

SN - 0011-4642

VL - 66

SP - 895

EP - 911

JO - Czechoslovak Mathematical Journal

JF - Czechoslovak Mathematical Journal

IS - 3

ER -